A typical tire removal apparatus which operates by pressing a pneumatic tire off the rim of a motor vehicle wheel comprises a pressing blade which is mounted pivotably in a substantially horizontal plane of pivotal movement on a machine frame structure, in particular of a tire fitting and removing machine. The pressing blade is operable to release the pneumatic tire from the rim of the wheel before the tire is totally removed from the wheel. In that procedure, the tire is pressed over the hump of the rim. In such a machine the pressing blade is mounted to the machine frame structure pivotably horizontally in a fixed plane. As rims and pneumatic tires for motor vehicle wheels are of different sizes which are generally between 12 inches and 20 inches, that apparatus suffers from the disadvantage that it is only in rare cases that the pressing blade can be fitted accurately at the same level as the wheel axis. This means that the procedure for pressing the tire off the rim cannot be implemented in the optimum fashion, thereby entailing the risk of damaging the tire, for example by virtue of causing crushing deformation thereof, as well as the rim. Furthermore it is frequently necessary for the pressing blade to be applied to the tire a plurality of times, so that as a result the procedure for pressing the tire off the rim takes up a considerable amount of time.